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Concept
Kakutani fixed-point theorem
Summary
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.
The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics.
Kakutani's theorem states:
Let S be a non-empty, compact and convex subset of some Euclidean space Rn.
Let φ: S → 2S be a set-valued function on S with the following properties:
φ has a closed graph;
φ(x) is non-empty and convex for all x ∈ S.
Then φ has a fixed point.
Set-valued function A set-valued function φ from the set X to the set Y is some rule that associates one or more points in Y with each point in X. Formally it can be seen just as an ordinary function from X to the power set of Y, written as φ: X → 2Y, such that φ(x) is non-empty for every . Some prefer the term correspondence, which is used to refer to a function that for each input may return many outputs. Thus, each element of the domain corresponds to a subset of one or more elements of the range.
Closed graph A set-valued function φ: X → 2Y is said to have a closed graph if the set {(x,y) | y ∈ φ(x)} is a closed subset of X × Y in the product topology i.e. for all sequences and such that , and for all , we have .
Fixed point Let φ: X → 2X be a set-valued function. Then a ∈ X is a fixed point of φ if a ∈ φ(a).
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